Working Paper 96-45

Statistics and Econometrics Series 16

July, 1996

Departamento de Economía

Universidad Carlos III de Madrid

Calle Madrid, 126

28903 Getafe (Spain)

Fax (341) 624-98-75

SYMMETRICALLy NORMALIZED INSTRUMENT AL-VARIABLE

ESTIMATION USING PANEL DATA

César Alonso-Borrego * and Manuel Arellano ••

Abstract ________________________________

In this paper we discuss the estimation of panel data models with sequential moment restrictions

using symmetrically normalized GMM estimators. These estimators are asymptotically equivalent

to standard GMM but are invariant to normalization and tend to have a smaller finite sample bias.

They also have a very different behaviour compared to standard GMM when the instruments are

poor. We study the properties of SN-GMM estimators in relation to GMM, minimum distance and

pseudo maximum likelihood estimators for various versions of the AR(1) model with individual

effects by mean of simulations. The emphasis is not in assessing the value of enforcing particular

restrictions in the model; rather, we wish to evaluate the effects in small samples of using

alternative estimating criteria that produce asymptotically equivalent estimators for fixed T and

large N. Finally, as an empírical illustration, we estimate by SN-GMM employment and wage

equations using panels of UK and Spanish firms.

Keywords: Panel data, instrumental variables, symmetric normalization, autoregressive models,

employment equations.

• Departamento de Economía, Departamento de Estadística y Econometría de la Universidad

Carlos III de Madrid .•• CEMFI, Madrid

We thank Richard Blundell, Gary Chamberlain, Guido Imbens, Whitney Newey, Enrique Sentana,

Jim Stock an seminar audiences at Harvard, Princeton and Northwestern for useful comments. An

earlier version of this paper was presented at the ESRC Econometric Study Group Annual

Conference, Bristol, July 1994, and at the Econometric Society European Meeting in Maastricht,

August 1994.

1. Introduction

In this paper we present instrumental variable estimators of

panel data models with predetermined variables subject to a symmetric

normalization rule of the coefficients of the endogenous variables. We

also evaluate the performance of these techniques for first-order

autoregressive models with individual effects by mean of simulations.

Lastly, an empirical illustration is provided.

This work is motivated by a concern with the biases of ordinary

IV estimators when the instruments are poor. A linear panel data model

wl th predetermined variables, typically estlmated by IV techniques,

takes the form

E(Lly - Llx' <5 z .. z ) = O, (t=1, .. ,T; i=1, .. ,N).i t i t 11 i t

This formulation includes vector autoregressions and linear Euler

equations. The specification of the equation error in first­

differences reflects the fact that the analysis is conditional on an

unobservable individual effect. Since the number of instruments

increases with T, the model generates many overidentifying

restrictions even for moderate values of T. However, often the quality

of the instruments is poor given that it is usually difficult to

predict variables in first differences on the basis of past values of

other variables.

The weaker the correlation of the instruments with the endogenous

variables, the smaller the amount of information on the structural

parameters for a given sample size. However, as it is well documented

in the literature on the finite sample properties of simultaneous

1

equations estimators, the way in which this situation is reflected in

the distributions of 2SLS and LIML differs substantially, despite the

fact that both estimators have the same asymptotic distribution. While

the distribution of LIML is centred at the parameter value, 2SLS is

biased towards OLS, and in the completely unidentified case converges

to a random variable with the OLS probabili ty limit as its central

value. On the other hand, LIML has no finite moments regardless of the

sample size, and as a consequence its distribution has thicker tails

than that of 2SLS and a higher probability of extreme values (see

Phlllips (1983) for a good survey of the literature). As a result of

numerical comparisons of the two distributions involving median-bias,

interquartile ranges and rates of approach to normali ty, Anderson,

Kunitomo and Sawa (1982) conclude that LIML is to be strongly

preferred to 2SLS, particularly if the number of outside lnstruments

is large. Similar conclusions emerge from the results of asymptotic

approximations based on an increasing number of instruments as the

sample size tends to lnfini ty; under these sequences, LIML is a

conslstent estimator but 2SLS is inconslstent (cf. Kunitomo (1980),

Morimune (1983) ando more recently, Bekker (1994)).1 (In our contexto

these approximations would amount to allowlng T to increase to

inflnlty at a chosen rate as opposed to the standard flxed T, large N

asymptotics. )

Despite this favourable evidence. LIML has not been used as much

in applications as instrumental variables estimators. In the past,

LIML was at a disadvantage relative to 2SLS on computational grounds.

More fundamentally, applied econometric1ans have often regarded 2SLS

as a more "flexible" choice than LIML from the point of vlew of the

2

restrictions they were will1ng to impose on their models. In effect,

the IV techniques used for a panel data model wi th predetermined

instruments are not standard 2SLS estimators, since the model gives

rise to a system of equations (one for each time period) wi th a

different number of instruments available for each equation. Moreover,

concern with heteroskedasticity has lead to consider alternative GMM

estimators that use as weighting matrix more robust estimators of the

variances and covariances of the orthogonal1 ty condi tions (following

the work of Chamberlain (1982), Hansen (1982) and White (1982)).

In a recent paper, Hillier (1990) shows that the alternative

normalization rules adopted by LIML and 2SLS are at the root of their

different sampling behaviour. Indeed, Hill1er shows that the

symmetrically normalized 2SLS estimator (SN-2SLS) has essentially

similar properties to those of the LIML estimator. This result, which

motivates our focus on symmetrically normalized estimation, is

interesting because SN-2SLS, unlike LIML, is a GMM estimator based on

structural form orthogonality conditions and therefore it can be

readily extended to the nonstandard IV situations that are of interest

in panel data models wi th predetermined variables, while relying on

standard GMM asymptotic theory.

To illustrate the situation, let us consider a simple structural

equation with a single endogenous explanatory variable and a matrix of

instruments Z:

y = (3x + u (1.1)

Letting y and x be the OLS fitted values from the reduced form

3

.equations

y = Zn + v 1

(1. 2)X = Zr + v

2

the 2SLS est1mator of ~ 1s g1ven by

" = Cov(x:y ) Cov(x,y) == A~2SLS

Var(x) COV(X,X)

which is not invariant to normal1zation except 1n the just-identified

case. That 15, it differs from the indirect 2SLS estimator:

..." = Var(y) Cov(y.y)~I2SLS " Cov(y,x) Cov{y,x)

On the other hand, the SN-2SLS estimator is given by the orthogonal

regression of Y on x, which is invariant to normalization:

" ...

= Cov(x,y) Var(y)-I\.== -----;:~-~SN ... "

Var(x)-I\. Cov(y,x)

The stat1stic 1\. is the minimum eigenvalue of the covariance matrix of

y and x.

The three estimators have the same first-order asymptotic

distribution, but satisfy the inequality

4

Moreover, ~SN can be written as

COy (x+~ y, y)SN

~SN= A " "

Cov(x+~ y.x)SN

Therefore. 2SLS, I2SLS and SN can al! be interpreted as simple IV

estimators that use as instruments x,y and x + ~ y. respectively.SN

Symmetrically normalized 2SLS can also be given a straightforward

interpretation as a GMM or minimum distance estimator. which

highlights its relation to LIML. Indeed, both SN-2SLS and LIML are

least-squares estimators of the reduced form (1.2) imposing the over­

identifying restrictions n=~r. Let us define

(~ .1 ) = argmin [y-zr~l' (V-1®I) [y-zr~lv v x-Zr x-Zr

~.r

Concentrating r out of the LS criterion we obtain

~v = argmin ~

It turns out that LIML is -~ with V equal to the reduced form v

residual covariance matrix while SN-2SLS is ~v wi th V equal to an

5

'1

identity matrix (cf. Malinvaud (1970), Goldberger and Olkin (1971) and

Keller (1975», so that both LIML and SN-2SLS solve minimum eigenvalue

problems. In particular, SN-2SLS is a GMM estimator based on the unit­

length orthogonality conditions

Notice that in spite of V being a matrix scaling factor, the

asymptotic distributlon of ~ does not depend on the choice of V. This v

,.. is so because optimal MD estimators of ~ based on (n-1~,1-1) and on

,.. (n-1~) are asymptotically equivalent, due to the fact that the

limi ting distribution of opt1mal MD 1s invar1ant to transformations

and to the add1tion of unrestricted moments.

The paper is organized as follows. Section 2 begins with a

formulation of the SN-2SLS estimator and its relation to 2SLS and LIML

in the general context of a linear structural equation. Next, we

present two-step SN-GMM estimators and test statistics of over­

identifying restrictions for panel data models with predetermined

instruments. Section 3 studies the finite sample properties of SN-GMM

estimates in relation to ordinary GMM. minimum distance and pseudo

maximum likelihood estimators for various versions of the first-order

autoregress1ve model with individual effects. The objective is not to

assess the value of enforcing particular restrictions in the model,

but rather to evaluate the effects in small samples, by mean of

simulations, of using alternative asymptotically equlvalent estimators

for fixed T and large N. Section 4 re-estimates the employment

6

equations for a sample of UK firms reported by Arellano and Bond

(1991) using symmetrically normalized and indirect GMM estimators.

This section further illustrates the techniques by presenting SN-GMM

estimates and bootstrap confidence lntervals of employment and wage

vector autoregresslons from a larger panel of Spanlsh flrms. Flnally,

Section 5 contalns the conclusions of the paper.

2. The Symmetrically Normalized Instrumental-Variable Estimator

Preliminaries

We begin this section by providing explicit express10ns for 2SLS,

LIML and symmetrically normalized 2SLS estimators in order to

highlight the algebraic and statistical connections among the three

statistics.

Let us cons1der a standard linear structural equation

y = y ~ + z o + u =Xo + u. (2.1 )1 2 1

Also let Y=(y ,Y ) be the nx(l+p) matrix of observations of the1 2

endogenous variables, and let Z=(Z ,Z) be the nxk matr1x of 1 2

1nstruments, where Z is nxk ,Z 1s nxk , and k ~p.1 1 2 2 2

The two-stage least squares (2SLS) estimator of o 1s given by

o = argmin a'W'MWa (2.2)2SLS o

wlth W=(Y,Z), M=ZeZ'Z)-lZ' and a=(l.-~· ,-o')'. An expression for the 1

partition of o is given by2SLS

7

= argmin b'Y' (M-M )Yb = [Y' (M-M )Y ]-ly ' (M-M )y(32SLS 2 1 2 2 1 1(3 1

with b=(1, -(3' )' and M =Z (Z' Z) -1Z'. 1 1 1 1 1

Similarly, the LIML estimator is given by

a'W'MWa(3 = argmin " = [X' (M-i(I-M)/n)X]-IX' (M-i(I-M)/n)y (2.3)LIML 1(3 b'Qb

where A=min eigen[Y' (M-M )YQ"-1 ] and Q=Y' (I-M)Y/n, which can be1

partitioned in accordance with Y as

A

Notice that A~O. Equally,

b' Y' (M-M )Yb = argmin __~,,_1__ = [Y' (M-M )Y -ic ]-1 [Y' (M-M )y -i~ ]

(3LIML (3 b' Qb 2 1 2 22 2 1 1 21

We define the orthogonal or symmetrically normalized 2SLS

estimator (SN-2SLS) to be (see Keller (1975) and Hillier (1990»:

8

• • ••

• •

a'W'M'Wa

° = argmin ----;;-¡-;-- (2.4)SNM

° Let Wa =Yb +2 c =u denote equation (2.1) without imposing a

1

•normal1zatlon rule. With the normal1zatlon used by 2SLS a =a, while

with a symmetric normalization of the coefficients of the endogenous

variables a• =O+{3' (3) -1/2a. Thus 0SNM is the minimizer of a'• W' M'Wa•

subject to b 'b =1.

Minimizing the criterion (2.4) with respect to r we obtain a

concentrated criterion that only depends on {3. This gives us:

b' Y' (M-M )Yb1 = argmin ----Cb'""'''b-- = [y; (M-M )Y - ~Il-1y; (M-M )Y1

1 2 1{3

= (2'2 )-12, (y -y ~ )1 1 1 1 2 SNM

where A=min eigen[Y' (M-M )Yl. Notice that also A=min(a'W'M'Wa)/b'b and 1

that A~O. Equivalently,

(2.5)

where ~ = [~ ~l. A

In the just identified case, 2' (y -Xo )=0 which min1mizes the 1 2SLS

three criteria, so that A=A=O, with the result that 2SLS, LIML and SN­

2SLS coincide.

Both 0LIML and 0SNM are invariant to norma11zation while 02SLS is

noto 2 That is, if the equation 1s solved for an endogenous variable

other than Y1' contrary to the case with 2SLS, the indirect estimates

9

obtained from o or o coincide wi th .the direct SNM' or LIMLSNM LIML

estimates, respectively.3

The LIML estlmator can be regarded as a minimum distance or

generallzed nonlinear least squares estlmator based on the reduced

form (see Malinvaud (1970) and Goldberger and Olkin (1971)).

Similarly, the SN-2SLS estimator can be viewed as an ordinary

nonlinear least squares estimator. To see this, let the reduced form

of Y be

y = ZTI' + V. (2.6)

In view of the partition in Y, the (l+p)xk matrix of reduced form

coefficients can be partitioned as TI'={n , TI;). In addition, given the 1

structural equation we have

n~ = ~'TI2 + (o' ,0' ) (2.7)

so that TI is a function of ~, o and TI • We can consider NLS estimators 2

of o and TI that solve2

(o TI ) - argmin tr[V-1 (Y-ZTI' )' (Y-ZTl' )] (2.8)NLS' 2,NLS ­

for particular choices of V. This class of estimators was proposed by

Keller (1975). Since TI is not of direct interest we can obtain a 2

concentrated NLS criterion that only depends on o, which gives 0NLS as

the solutlon to

10

a'W'MWa es = argmin b' Vb . (2.9)NLS

Clearly, LIML is es NLS

with V=O whlle SN-2SLS is esNLS

with V=I.

The choice of V, provided it is assumed to be bounded in probability ...

or a nonstochastic matrix, leaves the asymptotic distribution of es NLS

unaffected and equal to that of the 2SLS estimator. This result is

similar to the one that establishes the equivalence between 2SLS and

3SLS in a system in which there is only one overidentified structural

equation.

Symmetrically normalized estimators are attractive alternatives

to 2SLS on at least three grounds. Firstly, they tend to have a

smaller finite sample bias than the 2SLS estimators. Hillier (1990)

shows that for the normal case with p=l SN-2SLS and LIML are

"spherically unbiased" in finite sainples. 4 However, 2SLS does not have

this property.

Secondly, the concentration of the densities of the symmetrically

normalized estimators depends on the quality of the instruments. In

the completely unidentified case, as shown by Hillier, these

estimators have a uniform distribution on the unit circle. This is in

contrast with 2SLS which converges to the same llmlt as OLS and whose

distribution is determined exclusively by the normalization adopted.

When the instruments are poor, as well as when the number of

instruments is large relative to the sample size, 2SLS tends to

provide results that are biased in the direction of OLS and also large

discrepancies between "direct" and "indirect" 2SLS when using

different normalizations. This situation has been stressed in a number

of recent papers (Bekker (1994), Bound, Jaeger and Baker (1995»,

11

Staiger and stock (1994) and Angrist and Krueger (1995) amongst

others). In contrast, with poor instruments the distributions of LIML

and SN-2SLS accurately reproduce the fact that the information on the

structural parameters is very small.

Thirdly, they are invariant to normalization. SN-2SLS shares

these properties in common with LIML; however, one further advantage

of SN-2SLS in relation to LIML, is that it is a generalized method of

moments estimator based on structural form moment conditions and

therefore it can be easily extended to distribution free environments

and robust statlstlcs. In particular, i t is well sui ted for

application to nonstandard instrumental-variable problems such as

those that arise in the context of dynamic and error-in-variables

models for panel data.

As the previous discussion reveals, both LIML and SN-2SLS are GMM

estimators of o solved jointly with TI and based on the vector of the 2

reduced form orthogonality conditions:

(2.10)

where TI is a function of o and TI (both GMM estimators use a weighting2

matrix of the form (V®Z' Z) -1 wi th v=o. " for LIML and V=I for SN-2SLS).

However, SN-2SLS is also a GMM estimator of o based on the structural

form orthogonality conditions:

(2.11)

12

(In the last two expressions, z, y, y and Xl refer to the i-th 1 1 11

rows of 2, Y, Y1 and X respectively.)

There is one disadvantage, however, of SN-2SLS relative to the

other estimators.< In general, the results are not independent of the

units in which the variables are measured, so that a sensible choice

of the units of scale may be of sorne importance. 5

One further useful perspective on SN-2SLS can be obtained by

regarding it as a simple IV estimator. The statistic h can be written

as

" h = y~ (M-M ) (Yl-XoSNM)

1

Substituting this express ion in the formula for the estimator we

obtain

(2.12)

where

" 2 = X + (M-M1)Y1o~NMd

which reduces to Z=X+y o' if all the variables in X are endogenous.1 SNM

Remark that for 2SLS we have Z = X, and more generally for the j-th

indirect 2SLS estimator obtained by normalizing to unity the

coefficient on the j-th column of Y, we have 2=W(j) , where W(j) "­

coincides w1th W=(Y. 2 1

) except for the j-th column of Y which is

omitted.

13

Models lor Panel Data

We consider a model with individual effects for panel data given

by

= x' Ó + U (t=1 •...• T; i=1 •...• N) (2.13)1t lt

u = 1) + v 1t lIt

The model specifies sequential moment conditions of the form

E(vlt I (2.14)

were z;=(zl~ '" zl~)' is a vector of instrumental variables.

Thus. this setting is sufficiently general to cover models with

strictly exogenous. predetermined and endogenous explanatory

variables. We assume that

i=1 •... N} is a random sample (iid) of size N.

Estimation will be based on a sequence of orthogonality

conditions of the form

(t=1 •...• T-1) (2.15)

where starred variables denote forward differences or orthogonal

deviations of the original variables (e.g. y;t=Yl(t+1)-Ylt)'

It is convenient to rewrite the transformed model in the form

y. = X·ó + u· 1 1 1

14

• • • where y -(y )' etc1- 11 Y1 (T-1) , •

The mx1 parameter vector o 1s usually est1mated by GMM lead1ng to

est1mators of the form (see Holtz-Eak1n, Newey and Rosen (1988),

Arellano and Bond (1991), Chamberla1n (1992), Arellano and Bover

(1995), and Ahn and Schmidt (1995) amongst others):

(2.16)

where y.=(y.. ' ... y.')', X·=(X·' ... X·')' and 2=(2i ... 2N)'. 21 1s a (T­1 N 1 N

t1 )xq block diagonal matrix whose t-th block is Z1' and AN 1s chosen

such that it is a consistent estimate of the inverse of E(2'u·u·'2 J. I I 1 1

The standard robust choice is

AN = (~ 2'u·u·'2 )-1L.1 I I 1 1

where u· is a vector of residuals evaluated using some preliminary1

consistent estimate of o. Under very general regularity conditions

.fN'(5GHH-O) is asymptotically normal as N~ and T is fixed, and a

consistent estimator of the asymptotic variance of o is g1ven by GHH

(X·'2 A 2' X·)-l (2.17)N

Moreover, the Sargan or GMM statistic of overident1fylng

restrlctlons is glven by

s = u·, 2 A 2' u· ~ i N q-m

15

" where u". = y.- X·á .GHH

Turning to symmetrically normalized GMM (SNM) estimators of á,

let us consider a partition of X·=(X· X·) and a correspondingl' 2

parti tion of á=(á' á')' distinguishing between non-exogenous andl' 2

exogenous variables, such that the m columns of X· are linear 2 2

combinations of those of Z while the m columns of X· are noto 1 1

SNM is the GMM estimator of á based on the orthogonality

conditions

Z' (y·-X ·á -X ·á ) 1 E 1/1(1.1 ,á) = E 1 1 11 1 21 2 = O (2.18)

1 [ (1+á'á )1/2 1 1

Since E[I/1(w ,á)I/1'(w ,á)] = E(Z'u·u·'Z )/(l+á'á )=A l(l+á'á ) A1 1 1111 11 N 11' N

remains an optimal weighting matrix for the SNM estimator. Therefore,

(y·-X·á)'M·(y·-X·á)á = argmin (l+á'á ) (2.19)SNH

á 1 1

where M· = ZA Z'. Following our earlier discussion we obtain N

d'W·' (M·-M·)W·d1 1 2 1 1á = argmin (2.20)lSNH d'd

1 1á 1

(2.21)

where W· = (y. X·) d = (1 -á')' and M· = M·X·(X·'M·X·)-lX·'M·. So 1 '1' 1 '1 2 2222

that

16

= [X·' (M·-M·)X· - AIl-1 X·, (M·-M·)y· (2.22)1 2 1 1 2

wi th A = min eigen[W·' (M·-M·)W·]. A compact expression for o is . 1 2 1 SNH

given by

o = (X·'M·X· - AÓ)-l X·'M·y· (2.23)SNH

A A

Since O and O areasymptotically equivalent, Vado ) isGMM SNM GMH

also a consistent estímate of the asymptotic varlance of OSNH

A A

However, an alternatíve natural estímator of Vado ), suggested bySNM

theexpresslon above, is

A "

Vareo ) = (X·'M·X· - AÓ)-l (2.24)SNM

Moreover, since A is a minlmized optimal GMM crlterion it can be used

as an alternative test statistic of overidentifying restrictions. We

have the result

(1 + o' o )A ~ -l (2.25)lSNM lSNM q-m

which 1s asymptotically equivalent to the Sargan test.

The ex1sting evidence from Monte CarIo experiments and empirical

analysis point in the direct10n that, even for moderately large cross­

17

sectional sample sizes, ordinary GMM estimates and their standard

errors can be worryingly biased when the instruments are poor. This is

typically the case in the context of autoregressive models with

individual effects when the roots are close to unity or the

contribution of the permanent effect to the total variance is high. If

the desirable fini te sample properties of symmetrical1y normalized

estimators apply to these environments, o , Var(o ) and A could SNK SNM

provide a useful alternative to estimation and testing.

3. Experimental Comparisons with Alternative Estimators for First

Order Autoregressions with Random Effects

The purpose of this section is to study the finite sample

properties of the symmetrically normalized GMM estimators in relation

to ordinary GMM for varIous versions of the first-order autoregressive

model with Individual effects. The IV restrictions implled by these

models can also be represented as simple structures on the covariance

matrix of the data, and so we can also make comparisons with minimum

distance and pseudo maximum likelihood estImators of these covariance

structures. The emphasis is not in assessing the value of enforcing

particular restrIctions in the model, as done for example by Ahn and

Schmidt (1995) and Arellano and Bover (1995) for quadratic and

stationarity restrictions, respectively. Rather, we wish to evaluate

the effects in small samples of usIng alternative estimatIng criteria

that produce asymptotically equivalent estimators for fixed T and

large N. However, since we present results for three different sets of

moment restrictions, we shall also be able to make some comparisons

18

across models. We concentra te on a random effects AR(l) model because

of its simplicity and the fact that it is a case that has received a

great deal of attention in the literature.

Hodels and Estimators

Let us consider a random sample of individual time-series of size

T Y:=(Yl1, , .. 'Y1T)' (1=1"" ,N) with second-order moment matrix

E(//' )=Q={w }. We assume that the joint distribution of /1 and the 1 1 ts

unobservable time-invariant effect satisfies the following

assumption:

Assumption A

Ylt = l' + "Y + 'V> + V (t=2, ... , T) (3.1)1(t-1) "1 lt

(3.2)

Notice that since equation (3.1) includes a constant term, it is

not restrictive to assume that 11 has zero mean. However, in general1

TE(11 IYT ) will be a function of Y ' Moreover, the dependence between 11

1 1 1 1

and v is not restricted by Assumption A. Another remark is that lt

Assumption A does not rule out the possibility of conditional

heteroskedasticity, since E(v~tly~-1) need not coincide with ~~.

Following Arellano and Bond (1991), Assumption A implies (T-2)(T­

1)/2 linear moment restrictions of the form

19

(3.3)

These restrictions can also be represented as constraints on the

elements of n. Multiplying (3.1) by Yls for s<t, and taking

expectations gives:

w =aw +c (t=2, ... Ti s=l, ... ,t-l) (3.4)ts (t-l)s s

where e =E[y (r+ij )]. This means that, given Assumption A, the s 1s 1

T(T+l)/2 different elements of n can be writ ten as functions of the

2Txl parameter vector

We call this moment structure Model 1. Since the moment restrictions

in (3.3) are linear in a, they can be used as the basis for a linear

GMM estimator of the type discussed in the previous section.

The orthogonality conditions (3.3) are the only restrictions

implied by Assumption A on the second-order moments of the data. 7 In

particular, wi th T=3 the parameters (a, e ,e ) are just-identified as1 2

functions of the elements of n.

Model 1 is attractive because it is based on minimal assumptions.

However, we may be wllling to impose addi tional structure if this

conforms to a priori bellefs. One possibility is to assume that the

errors vare mean independent of the individual effect ij givenlt 1

Ylt-l . This situation gives rise to Assumption B.

20

Assumption B

(3.5)

Note that Assumption B is more restrictive than Assumption A. When

T~4, Assumption B implies the following additional T-3 moment

restrictions

In effect, we can write

E [ (y - 'Y - exy - ) (lIy - exlly )] = O'n 1t a 1 (t-l) "1 Ut-l) Ut-2)

and since E[('1+r¡ )lIv ]=0 the result follows. GMM estimators of ex 1 1 (t-l)

that exploi t these restrictions inaddi tion to those in (3.3) have

been considered by Ahn and Schmidt (1995). An alternative

representation of the restrictions in (3.6) is in terms of a recursion

of the coefficients c introduced in (3.5). Multiplying (3.1) byt

('1+r¡ ) and taking expectations gives:1

(t=2, ... ,T) (3.7)

where 4>=l+0'2=E[('1+r¡ )2], so that c ... c can be written in terms of r¡ 1 1 T-1

C and 4>. This gives rise to Model 2 in which Q depends on the (T+3)x11

parameter vector

21

Notice that with T=3 Assumption B does not imply further restrictions

in O with the result that a remains just identified relat1ve to the

second-order moments.

other forms of addit10nal structure that can be imposed are

various versions of mean or variance stationarity condit10ns.

Assumpt10n C. wh1ch requires the change in ylt

to be mean independent

of the individual effect T/ 1 •

1s a particularly useful mean

stationarity condition.

Assumption e

(t=2 •...• T) (3.8)

Notice that in combination with Assumption B. Assumption e

1mplies

= r + aE(y IT/) + T/1t-l 1 1

so that if E(Ylt lT/l) 1s constant it must be the case that

(r+T/ )/(1-0:) (3.9)1

and E (y 1t ) =r/ (1-0:) .

Relative to Assumption A and Model 1. Assumption e adds the

following (T-2) moment restrictions on o:

22

(t=3, ... ,T) (3.10)

whieh were proposed by Arellano and Bover (1995), who developed a

linear GMM estimator of a on the basis of (3.3) and (3.10).8 However,

relative to Model 2, Assumption e only adds one moment restrietion

whieh can be written as

(3.11)

In terms of the parameters e , the implieation of Assumption e is that t.

e = ... =e if we move from Model 1, or that e =</>/(1-a) if we move 1 T-l 1

from Model 2. This gives rise to Model 3 in whieh Q depends on the

(T+2)x1 parameter vector

Notiee that with T=3, a 1s overidentified under Assumption C.

The basie speeifieation can be restrieted further in various

ways. For example, we could consider time series homoskedasticity of

the form E(v2 )=0"2 for t=2, ... , T and stationari ty of the varianee of

lt.

the initial eonditions. The eombination of these assumptions with

Models 2 or 3 would give rise to additional models, some of which have

been discussed in detail in the paper by Ahn and Sehmidt (1995).

However, in the simulations we eoncentrate in Models 1, 2 and 3

beeause they embody the restrietions that have been found most useful

in applieations.

23

If .E(I/J (yT ,ex) ]=0 denotes the vector of orthogonality conditions j 1

available for Model j (j=l,2,3), the symmetrically normalized

estimators that we consider are the optimal GMM estimators based on

the restrictions E(I/J (y ,ex)/(1+ex2)1/2]=O, For example, the SNM j 1

estimator of ex for Model 1 is given by

b' A b 1 N o ex = ----= (3.12)

SNM,1 b' A b - A

1 N 1

-1~ -1r: - ­where b =N Lo Z'!J.y A =(N Z' !J.v bv' Z ) -1 o 1=1 1 1 ' N 1=1 1 1 1 1 '

A=min eigen(B'A B), !J.y =(by ., ,by)' ,N 1 13 lT

by (1-1) = (!J.y12' , .!J.yi(T-l»' and ZI is a (T-Z)x(T-Z) (T-1)/2 block

sdiagonal matrix whose sth block is given by Yl'

All three models can also be estimated by minimum distance (MD)

or by pseudo maximurn likelihood (PML) on the basis of the rnatrix of

A -1~ T Tsarnple second-order rnornents Q=N L. Y Y " and the representations as

1=1 1 1

covariance structures discussed aboye,

Optimal MD estirnators minimize a criterion of the form

(3.13)

where

A

mes) = vech[Q - Q(8)] = w - w(8)

and

v = N-1~ W w' - ww' N Ll=1 1 1

24

T T ~ ~

with w1=vech(Y1Yl') and w=vech(O).

These estimators have the same asymptotlc d1str1butlon as the

correspond1ng GMM and SNM est1mators. To see this for Model 1, not1ce

that

~

=H(a)[w-w(S)]1

where H1 (a) 1s a (T-l)(T-2)/2 x T(T+l)/2 seIection matrix that depends

on a. H (a) eIiminates (2T-l) moments which depend on the 2T 1

parameters contained in S. Taking into account that the limiting

distribution of optimal MD estimators is invariant to transformations

and to the addi tion of unrestricted moments, the asymptotic

equivaIence between GMM and MD follows.

Turning to PML estimators, one possibiI1ty, and the one that we

simuIate, is to minimize the criterion

(3.14)

subject to 0(8»0. 9 The first-order conditions for this PMLE are given

by:

where K is a 0-1 matr1x such that K vech(O)=vec(O). It turns out that

this PMLE 1s asymptotically equivaIent to the MD est1mator that uses

~-1 ~-1K' (Q ®O )K as the welghting matrix. Under our Monte CarIo deslgn

25

A_1 A_1 -1 plim[K' (Q ®Q )K-V ] = O. However, in other environments, such as

N

non-normal or noncentred data, this PMLE would be strictly less

efficient asymptotically that the optimal MDE.

An alternative PMLE which is always asymptotically equivalent to

the opt1mal MDE, minimizes

c·Ce) = log detCN-l~ [w - wCe)] [w - wCe)]') (3.15) m 1=1 1 1

S1nce the minimizer of c·Ce) is equivalent to the iterated MD and it m

can be expected to be very similar to the MO, 1t was not included in

the simulations.

Monte Carlo Results

We are particularly interested to analyze the behaviour of the

estimators in relat10n w1th the quality of the instruments. In Model 1

the quality of the instruments basically depends on the values of ex

2 2and r=O' r¡

/0' . To illustrate the situation. notice that under

stat10narity the correlat1on between ~y and y 15 g1ven by t-l t-2

p = - C1 - ex) [2 C1 - ex + C1 + ex) r ) ] -112

which produces the values

p ex =0.5 ex = 0.8

r = O -0.50 -0.32

r = 0.2 -0.39 -0.19

r = 1 -0.25 -0.10

26

11;·····-·'1

For this reason, we exclude from the simulations models w1th small

values of ex, which can be expected to perform relatlvely well. We

consider cases with ex=0.5, 0.8, ~ 2 =0, 0.2, 1, T=4, 7 and N=100. The 1)

variance of the random error ~2 is kept equal to unity for all cases.

For each experiment we generated 1000 samples of N independent

observatlons of (y , .... , y ) from the process11 iT

Y =exy +"" +v (t=2, ... ,T)1t 1 (t-l) "1 it

with v = (v •.... v )' - N(O.1) and 1) - N(O.~2 ) independent of v .

1 11 iT i 1) 1

Table 1 reports sample medians, percentage biases, interquartile

ranges and median absolute errors for pseudo maximum likelihood (ML),

minimum distance (MO). two-step GMM and symmetrically normalized two­

1. 10step GMM (SNM) estimators for Model The weighting matrices of GMM

and SNM are based on optimal one-step GMM residuals as described in

Arellano and Bond (1991). In almost every case, SNM is the estimator

with the smallest bias and the largest dispersion. When ~2=0 all 1)

estimators perform very well. although ML and MO have a smaller

interquartile range than GMM and SNM. a difference which is specially

noticeable for T=4 (with ~2=0 and ex=0.8 the interquartile range of ML 1)

or MO is about three times smaller than that of the ordinary or the

symmetrically normalized GMM estimators). When ~2=0.2 or 1, the 1)

differences in the distributions of GMM and SNM become apparent: the

higher ~2 or ex. the larger the negative bias of GMM for a given T, 1)

whereas SNM remains essentially median unbiased. SNM always has a

27

larger interquartile range than GMM, but the differences are small

except in the almost unidentified cases (with 0:=0.8 and T=4). The

median absolute errors of GMM and SNM estimates are of a very similar

magnitude. although those for GMM tend to be smaller than those for

SNM with T=4 and larger with T=7. With T=7. Table 1 clearly indicates

that when N=100 there is information in the data to estimate o: with

sufficient precision but that, contrary to SNM. GMM estimates may

still be substantially biased. As far as median bias is concerned, ML

and MD are practically unbiased when 0:=0.5, but exhibit sorne

worryingly large biases when u 2 is not zero and 0:=0.8. l)

The evldence from Table 1 suggests that Hillier's basic results

for ordinary and symmetrically normalized 2SLS estimators may have a

wider applicability. In effect, GMM and SNM, unlike 2SLS. are not only

functions of the second moments of the data but also of the fourth

order moments that enter the weighting matrix of the moment

condi tions.

Model 1 is the leading case from the point of view that

instrumental-variable estimatdrs of structural equations with

predetermined instruments tend to rely on orthogonal ity conditions

that are similar to those in Model 1.

Table 2 reports sorne results for Model 2 that exploits the (T-3)

quadratic restrictions given in (3.6) in addition to the linear ones

in (3.3). GMM and SNM are asymptotically efficient two-step GMM

estimates whose weighting matrix has been calculated using one-step

GMM residuals based on the same orthogonality conditions but weighted

by an identity matrix. We found that the results are sensitive to the

choice of residuals used by the two-step estimates. Unfortunately, in

28

this case, in contrast with the situation for Model 1, there does not

seem to be a "natural" choice of one-step GMM estimator that would be

asymptotically efficient under classical errors. Another problem is

that now GMM is not a linear IV estimator, so that the Justification

for an estimator based on the downweighted restrictions

2 -1/2E[(l+o::) Vl (Yl'O::)]=O becomes dubious. We also tded a version of J

SNM that only applied the symmetric normalization to the linear

orthogonality conditions with very similar results.

In Table 2, ML is, except in two cases, the estimator with the

smallest interquartile range and often the one with the smallest bias,

with MD trailing ML fairly closely. In drawing comparisons among the

estimators, it should be taken into account that the simulated data is

normally distributed, so that ML is implicitly using optimally

weighted moments with less sampling variability than the methods that

rely on higher order moments. On the other hand, ML and MD are subject

to the inequality restriction 10::1<1 while GMM and SNM are noto We

experimented with versions of GMM and SNM subject to 10::1<1 but this

did not alter qualitatively the results. Turning to the comparison

between GMM and SNM, SNM always has a smaller median bias than GMM,

al though SNM can also be substantially biased as in the experiment

with 0::=0.8, T=7 and ~ 2 =1. Nevertheless, we insist that these results 11

are sensitive to the choice of one-step residuals and further

investigation is required.

Table 3 presents the results for Model 3 which makes use of the

restrictions derived from Assumptions B and C. This model incorporates

the orthogonallty conditions from Model 2. However, by adding the

stationarity restrictions the entire list of moment conditions admits

29

a linear representation (cf. Ahn and Schmidt (1995)), so that GMM in

Table 3 is a linear IV estimator (as proposed by Arellano and Bover

(1995)). AII the estimators in this Table exhibit small median biases

and dispersions, al though, as in TabIe 2, the comparisons favour ML

and MD. The differences between GMM and SNM are small in most cases

without a clear pattern in the relation, except for the fact that on

average SNM estimates are always higher than the GMM estimates.

Both GMM and SNM are two-step estimators based on one-step GMM

residuals that use all the orthogonality conditions from Model 3, and

the inverse of the second moments of the instruments as the weighting

matrix. This one-step estimator is not asymptotically efficient, not

even under classical errors. Moreover, the results for GMM and SNM in

Table 3 are also sensi tive to the choice of one-step residuals. To

illustrate the situation, Table 4 reports results for GMM and SNM

estimates based on both one-step GMM residuals from Model 1 and one­

step residuals from Model 3, but using an identity as the weighting

matrix. As an extreme example, the median absolute error of GMM or SNM

in Table 3 can be seen to be haIf of the size of that of GMMb or SNMb

in Table A.1 for "=0.8, T=4 and q 2 =1. As one would expect, the impact1}

of using Model 1 residuals is more important when Model 1 estimates

are highly imprecise. These results suggest that an iterated GMM

estimator may often have very different finite sample properties

relative to a two-step estimator.

Finally, it is possible to make comparisons across tables. In

general, the interquartile ranges become smaller if we move from Table

1 to TabIe 2 and TabIe 3. The efficiency gains are particularly

important in the cases wi th "=0.8 and q2=0. 2 or 1. The gains from 1}

30

enforcing stationar1ty restrictions are always substantial for al! the

estlmators. A puzzling result is that for sorne experiments the ML and

MD estimates of Model 2 have a larger lnterquartile range than the

correspondlng estlmates for Model 1. However, this result may be

related to problems of nonconvergence that we experienced for sorne of

the replications for ML and MD in Model 2.

We have also investigated the flnite sample dlstributions of the

standardized GMM and SNM "t-statistics" for Model 1 of the form

A-l/2 " t = v (a - al (3.16)

GMM,1 GMM,l GMH,I

t SNH,l

= "-1/2 .. v (a

SNH,l SNH,l - a) (3.17)

where is as defined in

-expression but with A replaced

variances are given by:

(3.12) and

by zero. The

has

estimated

a similar

asymptotic

A

V GHH,l

= 1I(b' A b )1 N 1

v = 1/(b' A b -~l SNH,l 1 N 1

Both t and tare asymptotically N(O,l). Since the usual GMM,l SNK,1

tests of hypotheses and confldence intervals rely on thls

approximation, lt is useful to check the accuracy of the approximatlon

for the sample slzes and parameter values consldered aboye.

Table 5 reports finite sample quantlles of the t-statistics based

on 10,000 replicatlons. We use a larger number of replications because

31

in this case the 0.9 and 0.95 quantiles in the upper tail of the

distribution are of special interest. The median shows that the

distributions of the GMM t-statistics are shifted to the left, w1th

the absolute value of the shift increasing wi th ex, fT and T. In 1)

contrast, the distributions of the SNM t-statistics are centered at

values very close to zero. Turning to the 0.9 and 0.95 quantiles, when

T=4 the differences with the corresponding N(O.l) quantiles are always

smaller for the SNM t-statistics than for the GMM, sometimes by a wide

margino When T=7, the normal approximation worsens for both

estimators. In that case, however, the upper-tail GMM quantiles tend

to be closer to the normal values than those from the SNM t­

statistics.

4. Empirical Illustrations

Our first illustration of the previous methods proceeds by re­

estimating the employment equations presented by Arellano and Bond

(1991) using symmetrically normalized and indirect GMM estimators.

The Arellano-Bond dataset consists on an unbalanced panel of 140

quoted companies from the UK, whose main activity 1s manufacturing and

for which seven, eight or nine continuous annual observations are

available for the period 1976-1984.

The models are all log-linear relationships between the number of

employees, the average real wage, the stock of capital, a measure of

industry output, lagged values of the previous variables, time dummies

and company effects. The reader 1s referredto the Arellano and Bond

article for a detailed description of the models and the data.

The first two panels of Table 6 contain the resul ts for two

32

different models estimated in flrst differences using instrumental

variables. Model A includes contemporaneous wage and capital

variables, which are treated as endogenous along with the first lag of

employment. In this model lagged sales and stocks are used as outside

instruments in addition to lags of the endogenous variables included

in the equation. Model B only includes lagged values of wages and

capital and it could be interpreted as an approximated Euler equation

for employment wi th quadratic adJustment costs. Columns labeled GMM

reproduce some of the resul ts obtained by Arellano and Bond. The SNM

estimates are calculated as described in Section 2, and for Model A

there is an additional column containing indirect GMM estimates that

were obtained by normalizing to unity the coefficient of

contemporaneous wages. Fl na lly • the third panel of Table 6 presents

GMM and SNM estimates of some simple second-order autoregressive

models for employment with and without the inclusion of lagged wages.

As Table 6 shows, SNM and indirect GMM estimates are far apart

from the direct GMM estimates. These results uncover the fact that the

GMM estimates from the dataset of UK flrms are probably much less

reliable than what their estimated asymptotic standard errors would

suggest. Interestingly, the SNM estimates of Model B are more

compatible with the Euler equation interpretation than the GMM

estimates. For example, in the Euler equation discussed by Arellano

and Bond the coefficient on n is given by (2+r) where r is the real t-l

discount rateo

Our second empirical illustration is based on a similar but

larger balanced panel of 738 Spanish manufacturing companies, for

which there are available annual observations for the period 1983-1990

33

,,;,,----,' -----,---------¡----------¡----¡r------------­

(see the AppendIx for a descrIption of these data). We cónsIder a

bIvarIate V!\R model for the logarithms of employment and wages. The

employment equation contalns both lagged employment and lagged wages,

whIle the wage equatIon only Includes its own lags. ThIs model can be

regarded as the reduced form of an intertemporal model of employment

determInation under rational expectations (see Sargent (978». To

obtain the reduced form, an !\R(2) process for log wages is assumed,

and the Euler equation in the log of employment for the optimum

contlngency plans is solved.

Table 7 presents GMM and SNM estimates of the two equations,

flrstly using only lagged variables in levels as instruments for

equations in flrst-differences (the baslc set of moment conditions

that we called "Model 1"), and secondly adding lagged variables in

first-differences as instruments for equations in levels (that is,

including the stationarity restrictions of "Model 3"). For Model 1 we

also report estimates of a univariate !\R(2) process for employment.

In addition to asymptotic confidence intervals, we calculated 95

percent semiparametric bootstrap confidence intervals based on 1000

replications from the empirical distribution function of the data

subject to the moment restrictions (cf. Back and Brown (1993».

Following Brown and Newey (1992) we drew the bootstrap samples from

the mass-point distribution that estimated the probability of the i-th

observation as

p = 1/(1+l'W(y ,9»N1 1

where

34

...,._~-_.~--------------¡---------------------------

i

... 1 N '" 2t = argmin -N L 10g[t+t'ljJ(y,S)]1=1 1

and ljJ(Yt'S) is the vector of orthogonality conditions for observation

evaluated at the appropriate parameter estimates.

rabIe 7 contains some interesting results. GMM and SNM estimates

of Model 1 are still different from each other but by a smaller margin

than the corresponding estimates for the UK panel. The difference

becomes even smaller for the univarlate employment estlmates that are

based on half the number of moments used for the estimates in the

first two columns. On the other hand, the estimates of Model 3 appear

to be more precise, presumably because the additional orthogonality

conditions are highly informative. In this case, GMM and SNM estimates

provide very similar results. However, the Sargan statistics indicate

a clear reJection of the stationarity restrictions in both the

employment and the wage equations. It is also noticeable that although

bootstrap confidence intervals are always larger than the asymptotic

confldence intervals, the differences between the two are generally

small.

\ole re-estimated Model 1 with a random subsample of 200 firms.

which is similar to the size of the UK sample. Interestingly. the

results (reported in rabIe 8) are closer to the UK results for similar

specifications than those based on the full Spanish sample. In

particular, the SNM estimates of the AR(2) model for employment are

remarkably stable over the three datasets whlle standard GMM estimates

would be seriously downward biased in the smaller samples. Moreover,

the discrepancies between asymptotic and bootstrap confidence

35

intervals in the random subsample were greater than in the full

sample. 11

Finally, we simulated data as clase as possible to the AR(2)

employment equation, to see if the findings that we obtained with the

subsample of 200 companies were substantiated in the Monte CarIo

simulations. Random errors and individual effects were generated from

independent normal distributions with variances equal to the values

estimated from the SNM residuals of the full Spanish sample. Since the

estimated time effects showed very little variability, the constant

was set to a common value for all periods given by the average

estimated time effect in levels, although the estimates in the

simulations included time dummies. As a consequence the model was

stationary, and we generated (and discarded) 100 preliminary

observations for each individual to minimize the impact of ini tial

conditions. The results are reported in Table 9, and confirm the

impression conveyed by the real data. The SNM estimates are almost

median unbiased, but GMM shows large downward biases, specially when

N=200. A comparison in terms of median absolute errors also favours

SNM for both sample sizes and parameter estimates. Lastly, looking at

the quantiles of the t-ratios shown in the lower panel of Table 9, it

appears that the N(O,l) approximation is reasonable for the SNM t­

ratios but not for the GMM t-ratios.

36

5. Conclusions

It has long been established that the lack of finite sample bias

is an important advantage of LIML estimators of structural equations

over 2SLS, which by contrast have thinner tails than LIML. The bias of

2SLS towards OLS can be specially worrying when the instruments are

"poor" and/or the degree of overidentification is l~rge. In practice,

this means that while LIML is invariant to normalization, often a 2SLS

regression of y on x provides results that are fairly different from

those of the (inverted) 2SLS regression of x on y, despite being

asymptotically equivalent estimators. However, LIML has not been used

much in applications. The reasons for this include a computational

disadvantage over 2SLS, concerns with outliers, the fact that 2SLS can

be more easily accommodated into the GMM framework, and we suspect

that sometimes the use of an implicit prior that favored closeness to

OLS when structural coefficients were poorly identified.

There has recently been a renewed interest in the finite sample

properties of GMM estimators in various time series and cross­

sectional contexts. Several papers have emphasized the role of

estimated weighting matrices for the properties of the estimators in

small samples, and a number of alternative methods have been

considered (eg. Altonji and Segal (1994), Hansen, Heaton and Varon

(1995), Angrist, Imbens and Krueger (1995) or Imbens (1995). In

contrast, in this paper we have focused on the role of normalization

rules for the finite sample properties of GMM estimators that make use

of standard two-step weighting matrices. Our work is motivated by the

results in Hillier (1990), who argued that the alternative

normalization rules adopted by LIML and 2SLS are at the basis of their

37

d1fferent sampling behaviour. Hillier showed that a symmetr1cally

normal1zed 2SLS has similar finite sample propert1es to those of LIML.

Th1s resul t 1s interestlng because, unlike LIML, SN-2SLS 1s a GMM

est1mator based on structural form moment cond1t1ons and therefore 1t

can be easlly extended to distr1butlon free env1ronments and robust

statlstics.

In particular, SN-2SLS 1s well sulted for appl1catlon to the

nonstandard IV s1tuations that arise in panel data models with

predetermined variables, which are the models of interest in this

papero These models are typically est1mated in first-differences using

all the avallable lags as instruments. Usual1y, there is a large

number of instruments avallable, but of poor quality since they tend

to be only weakly correlated wl th the first-differenced endogenous

variables that appear in the equation.

In this paper we have presented SN-GMM estimators for dynamic

panel data models that are asymptotically equivalent to ordinary

optimal GMM estimators. We have also showed how a byproduct of the

estimation is a test statistic of overidentifying restrictions, based

on a minimum eigenvalue calculatlon.

We have reported Monte CarIo evidence on the performance of GMM

and SN-GMM est1mates for a flrst-order autoregress1ve model with

individual effects. For this model we have considered three

alternative sets of moment conditions as discussed by Arellano and

Bond (1991), Ahn and Schmldt (1995), and Arellano and Bover (1995).

Since for these models, the IV restr1ct1qns can be expressed as

stralghtforward structures on the data covariance matrlx, using these

representations we have also calculated MD and QML estimates for

38

comparisons with the IV estimates. Our findings suggest that Hillier's

basic results may have a wider applicability. In most cases, SN-GMM is

the estimator wi th the smallest median bias, and the one wi th the

largest interquartile range. However, the differences in dispersion

with ordinary GMM are small except in the almost unidentified cases.

Finally, as an empirical illustration, we havereported estimates

of employment and wage equations from UK and Spanish firm panels. The

results show that GMM estimates from the (smaller) UK panel can be

very unreliable when the degree of overidentification is large. The

resul ts from the (larger) Spanish panel produce a closer agreement

between ordinary and symmetrically normalized GMM estimates, although

there is evidence that there can still be serious biases in GMM

estimates. Some of these results are confirmed by simulating data as

close as possible to the empirical data. Moment restricted bootstrap

confidence intervals show that asymptotic confidence intervals are

often over-optimistic, and Sargan tests consistently reject the

restrictions implied by the stationarity of initial conditions.

39

Footnotes

1. Split sample or jackknife IV estlmators, however, arealso conslstent when the number of lnstruments tends to lnflnity (cL Angrlst and Krueger (1995) and Angrlst, Imbens and Krueger (1995»).

2. Empirical likellhood estlmators of the type considered by Qin and Lawless (1994) and Imbens (1995) w111 also be lnvariant to normalization due to the invariance property of ML estimators.

3. Notlce that if the only explanatory exogenous variable in the

equation is a constant term, o coincides wi th the orthogonalSNM

regression on the fitted values Y (cf. Malinvaud (1970) and Anderson (1976»).

.... ...." 1/24. Meaning that the density of o: = b/(b' b) deflned on the unit

circle is symmetric about the true points ±o:=±b/(b'b)1/2 having modes at ±o:.

5. This problem does not arise in the autoregressive panel data models discussed below, since in that case the SN-GMM estimator is invariant to units and to normalization.

6. If no columns of X· are perfectly predictable from Z, or if the entire vector of coefficients is normalized to unity, then Á = I and

A=min eigen(W·'M·W·), with W· = (y. ,X·).

7. However, they are not the only restrictions available since (3.2)

also impl1es that nonlinear functlons of y~-2 are uncorrelated with

ÁV . The semiparametric efficiency bound for this model can be lt

obtalned from the results in Chamberlain (1992). One reason why estimators based on (3.3) may not be fully efficient asymptotically is

that the dependence between ~ and y T may be nonlinear. Another reason i i

would be unaccounted conditional heteroskedasticity.

8. Notice that the (T-2) restrictions in (3.10) can also be written as

(t=3, ... ,T)

For example, we have the identity

where u =y -o:yiT iT lIT-t)'

40

9. In all cases, optlmlzatlon wlth respect to o: was conducted over the range 10:1 <1. Thls was achleved uslng the reparameterlzatlon

0:=2P/{1+p2) .

10. Means and standard devlations are not reported slnce the symmetrlcally normalized estimators, in common with LIML. can be expected to have lnfinite moments.

11. Bootstrap standard errors for the UK unbalanced panel were not calculated, since they would depend on a nontrivial specification of the empirical distribution function for the unbalanced observations.

41

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Goldberger, A.S. and 1. Olkin (1971), "A Minimum-Distance Interpretation

42

-1111------------------,-------;------:----:--,---------- ­

of LImIted-Informatlon EstImatIon", Econometrlca, 39, 635-639.

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43

Data Appendix

The Spanish dataset is a balanced panel of 738 manufacturing companies recorded in the database of the Bank of Spain's Central Balance Sheet Office from 1983 to 1990. This survey contains information on firm's balance sheets and other complementary information, including data on employment and total wage bill. This survey started in 1982 with the collection of data from large companies with a tendency in subsequent years towards the addition of smaller companies. The database includes both quoted and non quoted firms. The manufacturing firms included in this data set represent more than 40% of the Spanish value added in manufacturing in 1985.

We selected firms reporting information during the whole period 1983-1990 that fulfilled several coherency condHions. Al! companies with negative values for net worth, capital stock. accumulated depreciation. accounting depreciation, labour costs, employment, sales, output or those whose book value of capi tal stock jumped by a factor greater than 3 from one year to the next, were dropped from the sample. Finally, we concentrated on non-energy, manufacturing companies with a public share lower than 50 percent.

Variable construction

Employment

Number of employees is dissagregated into permanent employees (those wi th long-term contracts) and temporary employees (those wi th short-term contracts). Total employment is calculated as the number of permanent employees, plus the average annual number of temporary employees (number of temporary employees during the year times the average number of weeks worked by temporary employees divided by 52).

Real wage

The measure of the firm's annual average labour costs per employee 1s computed as the ratio of total wages and salaries (in million Spanish pesetas) to total number of employees. This measure was deflated using Retail Price Indices for each of the subsectors of the manufacturing industry. (Source: Spain's Institute of National Statistics.)

Descriptive statistics

Mean Median Std. deviation Minimum Maximum

Employment 310.4 124.0 702.4 10.0 11004.0

Real Wage 1. 86 1. 75 0.67 0.32 6.66

44

rabie 1

l\Iodel 1: linear restrictiom

a=0.5 a=0.8

ML :MI) GMM SNM ML :MI) GMM SNM

T=4 <TTI=0

median 0.50 0.51 0.49 0.50 0.79 0.80 0.76 0.80 % bias 0.1 2.1 2.1 0.2 1.5 0.0 5.0 0.3

iqr 0.11 0.12 0.19 0.19 0.10 0.10 0.28 0.30 ¡q80 0.22 0.23 0.36 0.38 0.20 0.21 0.57 0.61 mae 0.05 0.06 0.09 0.09 0.05 0.05 0.15 0.15

<TTI= 0.2 median 0.50 0.51 0.47 0.49 0.69 0.71 0.65 0.76 % bias 0.1 1.3 6.4 1.8 13.7 n.3 18.7 4.5

iqr 0.19 0.20 0.24 0.25 0.28 0.28 0.47 0.55 iq80 0.36 0.39 0.47 0.50 0.54 0.58 0.94 1.30 mae 0.09 0.10 0.12 0.13 0.12 0.11 0.27 0.27

<TTI= 1 median 0.47 0.49 0.44 0.47 0.65 0.65 0.46 0.65 % bias 5.5 2.2 12.8 5.3 19.1 19.1 42.6 18.1

iqr 0.32 0.32 0.35 0.38 0.47 0.48 0.68 0.99 iq80 0.54 0.56 0.72 0.80 0.90 0.94 1.36 2.59 mae 0.15 0.16 0.18 0.19 0.18 0.18 0.43 0.51

T=7 <T1]=0

median 0.50 0.51 0.48 0.50 0.80 0.81 0.75 0.79 % bias 0.2 2.0 4.1 0.1 0.5 1.4 5.7 0.8

iqr 0.08 0.09 0.10 0.10 0.08 0.10 0.13 0.13 iq80 0.14 0.17 0.19 0.19 0.15 0.17 0.24 0.25 mae 0.04 0.04 0.05 0.05 0.04 0.05 0.07 0.07

<TTI= 0.2 median 0.50 0.50 0.47 0.50 0.74 0.74 0.69 0.79 % bias 0.3 0.1 6.2 0.5 7.7 7.8 13.7 1.7

iqr 0.10 0.12 0.12 0.12 0.14 0.17 0.20 0.20 iq80 0.19 0.23 0.23 0.23 0.27 0.34 0.39 0.41 mae 0.05 0.06 0.06 0.06 0.08 0.09 0.13 0.10

<T1]= 1 median 0.50 0.50 0.45 0.49 0.72 0.71 0.59 0.77 % bias 0.6 0.2 9.8 1.4 10.6 11.1 25.9 3.9

iqr 0.14 0.15 0.14 0.15 0.19 0.22 0.27 0.28 iq80 0.26 0.29 0.28 0.30 0.37 0.46 0.53 0.59 mae 0.07 0.08 0.08 0.07 0.10 o.n 0.21 0.15

1,OO~ reRlications. N=100, 0',,2=1~. ., . .,% biaS glves the percentage median bias for all esbmates; lqr IS the 75th-25th mterquartJle range; iq80 is the 9Oth-lOth interquantile range; mae denotes the median absolute error.

Table 2

1\1ode12: linear and quadmtic restrictions

0.=0.5 0.=0.8

:MI., :MI) GMM SNM :MI., :MI) GMM SNM

T=4 0211=0

median 0.50 0.51 0049 0.50 0.73 0.74 0.75 0.80 % bias 004 1.1 3.0 0.6 8.5 7.2 6.7 0.1

iqr 0.18 0.19 0.17 0.18 0.19 0.19 0.24 0.27 iq80 0.33 0.34 0.34 0.36 0.35 0.37 0.50 0.53 rnae 0.09 0.10 0.09 0.09 0.10 0.10 0.13 0.13

0211= 0.2 median 0049 0.50 0048 0.51 0.70 0.72 0.71 0.78 % bias lA 0.3 3.3 lA 12.0 10.3 10.8 2.9

iqr 0.22 0.22 0.20 0.23 0.22 0.23 0.27 0.33 iq80 0.39 0041 0.41 0046 0040 0041 0.56 0.63 rnae 0.11 0.10 0.10 0.11 0.12 0.13 0.16 0.16

0211 = 1 median 0048 0049 0048 0.52 0.72 0.73 0.63 0.71 % bias 4.3 1.7 404 304 10.3 9.2 21.2 11.2

iqr 0.23 0.24 0.24 0.27 0.24 0.25 0.33 0.39 iq80 0046 0046 0049 0.57 0.44 0045 0.67 0.71 rnae 0.12 0.12 0.12 0.13 0.14 0.13 0.22 0.21

T=7 0211=0

median 0.50 0.50 0047 0049 0.79 0.80 0.74 0.78 % bias 0.2 1.0 5.0 1.6 1.6 0.1 7.3 2.9

iqr 0.08 0.10 0.09 0.09 0.11 0.13 0.12 0.13 iq80 0.16 0.20 0.17 0.18 0.20 0.24 0.23 0.24 rnae 0.04 0.05 0.05 0.05 0.06 0.06 0.08 0.07

0211= 0.2 median 0.50 0.50 0047 0049 0.78 0.78 0.68 0.72 % bias 0.3 0.6 6.5 2.6 2.9 204 14.8 lOA

iqr 0.09 0.11 0.10 0.10 0.11 0.13 0.15 0.16 iq80 0.16 0.21 0.19 0.20 0.22 0.25 0.32 0.35 rnae 0.04 0.05 0.06 0.05 0.06 0.07 0.13 0.11

0211= 1 median 0.50 0.51 0045 0047 0.78 0.78 0.55 0.59 % bias 0.1 lA 10.9 6.9 2.8 204 30.7 26.8

iqr 0.08 0.11 0.11 0.11 0.12 0.15 0.23 0.24 iq80 0.17 0.22 0.22 0.24 0.23 0.26 0047 0048 rnae 0.04 0.05 0.07 0.07 0.06 0.07 0.25 0.22

See Notes to Table l.

, ,, l' i

Table 3

Model 3: linear and stationarity restrictiom

a=0.5 a=0.8

NIL fv1D GNIM SNM NIL fv1D GMM SNM

T=4 d=O11

median 0.50 0.51 0.50 0.51 0.80 0.81 0.79 0.81 % bias 0.1 1.2 0.8 2.1 0.1 0.7 0.9 1.5

iqr 0.07 0.07 0.15 0.15 0.05 0.05 0.17 0.17 iq80 0.12 0.14 0.28 0.28 0.09 0.09 0.32 0.31 mae 0.03 0.03 0.07 0.07 0.02 0.02 0.08 0.08

d 11 = 0.2 median 0.50 0.51 0.50 0.51 0.80 0.81 0.79 0.82 % bias 0.5 1.8 0.9 2.7 0.3 1.3 0.7 2.7

iqr 0.16 0.19 0.17 0.17 0.19 0.21 0.20 0.19 iq80 0.30 0.33 0.31 0.32 0.35 0.36 0.37 0.36 mae 0.08 0.09 0.09 0.09 0.09 0.10 0.10 0.10

d=l11 median 0.50 0.51 0.52 0.54 0.79 0.82 0.85 0.87 % bias 0.2 2.3 3.1 8.5 1.3 2.1 5.7 9.2

iqr 0.20 0.21 0.19 0.20 0.21 0.22 0.19 0.18 iq80 0.36 0.39 0.36 0.37 0.40 0.40 0.38 0.38 mae 0.10 0.11 0.09 0.10 0.09 0.10 0.11 0.11

T=7 d=O11

median 0.50 0.51 0.49 0.50 0.80 0.80 0.78 0.80 % bias 0.1 1.2 2.9 0.1 0.1 0.4 3.0 0.5

iqr 0.05 0.06 0.08 0.08 0.03 0.04 0.09 0.08 iq80 0.09 0.11 0.15 0.15 0.06 0.08 0.17 0.16 mae 0.02 0.03 0.04 0.04 0.02 0.02 0.05 0.04

d 11 = 0.2 median 0.50 0.50 0.49 0.50 0.80 0.81 0.78 0.80 % bias 0.3 0.6 2.6 0.9 0.2 1.1 2.4 0.5

iqr 0.08 0.10 0.09 0.09 0.09 0.12 0.11 0.10 iq80 0.15 0.20 0.18 0.18 0.17 0.22 0.20 0.19 mae 0.04 0.05 0.05 0.05 0.05 0.06 0.05 0.05

d=l11 median 0.50 0.50 0.50 0.51 0.80 0.81 0.83 0.85 % bias 0.2 0.4 0.7 2.9 0.1 1.8 3.5 5.7

iqr 0.08 0.11 0.10 0.11 0.11 0.13 0.12 0.11 iq80 0.16 0.22 0.19 0.20 0.20 0.25 0.22 0.21 mae 0.04 0.05 0.05 0.05 0.06 0.07 0.07 0.07

See Notes to Table 1.

Iable 4

GMM and SNM estimates for :Model 3 witb a1temative residuals

0:=0.5 0:=0.8

GMMa SNMa GMMb SNMb GMIvJa SNMa GMMb SNMb

T=4 02=011

median 0.49 0.51 0.49 0.51 0.77 0.81 0.79 .0.81 % bias 2.1 1.1 1.2 1.8 3.2 1.4 1.2 1.4

iqr 0.16 0.16 0.14 0.15 0.18 0.18 0.18 0.17 iq80 0.30 0.31 0.28 0.28 0.33 0.34 0.34 0.33 rnae 0.08 0.08 0.07 0.07 0.09 0.09 0.09 0.09

0211=0.2 median 0.49 0.51 0.49 0.51 0.79 0.83 0.75 0.78 % bias 1.2 2.6 2.1 1.1 1.3 4.4 6.7 2.1

iqr 0.18 0.19 0.18 0.18 0.20 0.19 0.26 0.26 ¡q80 0.33 0.35 0.32 0.32 0.36 0.36 0.48 0.47 rnae 0.09 0.09 0.09 0.09 0.10 0.11 0.14 0.13

02=111 median 0.52 0.55 0.48 0.51 0.86 0.90 0.66 0.73 % bias 3.3 10.1 4.3 1.8 7.2 12.5 17.3 9.1

iqr 0.21 0.22 0.21 0.22 0.17 0.16 0.41 0.42 ¡q80 0.38 0.39 0.39 0.39 0.34 0.34 0.76 0.85 rnae 0.11 0.12 0.11 0.11 0.10 0.12 0.22 0.21

T=7 02=011

median 0.46 0.50 0.49 0.51 0.74 0.81 0.78 0.80 % bias 7.9 0.4 1.7 1.1 7.9 1.5 2.3 0.0

iqr 0.10 0.11 0.08 0.08 0.11 0.11 0.09 0.09 ¡q80 0.19 0.21 0.15 0.15 0.21 0.21 0.17 0.16 rnae 0.06 0.05 0.04 0.04 0.07 0.06 0.05 0.04

0211=0.2 median 0.46 0.51 0.49 0.51 0.76 0.84 0.75 0.78 % bias 8.3 1.5 1.9 1.3 5.5 4.8 6.3 3.1

iqr 0.11 0.12 0.09 0.09 0.12 0.12 0.13 0.13 iq80 0.21 0.22 0.17 0.18 0.23 0.23 0.24 0.23 rnae 0.06 0.06 0.05 0.04 0.06 0.07 0.07 0.06

02=11')

median 0.49 0.54 0.48 0.50 0.83 0.90 0.68 0.70 % bias 2.7 8.6 3.9 0.3 4.0 12.3 15.5 12.6

iqr 0.12 0.13 0.11 0.11 0.11 0.10 0.18 0.18 ¡q80 0.23 0.25 0.19 0.20 0.22 0.20 0.35 0.34 rnae 0.06 0.07 0.06 0.05 0.06 0.10 0.13 0.11

See Notes to Table 1. G!v1Ma and SNMa use GMM residuals from Model 3 with wei~tin~ identity matrix. GMMb and SNMb use optimal one-step GMM residuals from ode 1.

Jable 5

l\1odel 1: linear restrictions QJantiles of tite t-statistics

T=4 T=7

a=0.5 a=0.8 a=0.5 a=0.8

GNlM SNM GMM SNM GMM SNM GMM SNM

0\=0 0.05 -2.04 -1.94 -2.25 -2.07 -2.49 -2.20 -2.74 -2.18 0.10 -1.61 -1.51 -1.80 -1.57 -2.01 -1.70 -2.28 -1.74 0.25 -0.87 -0.77 -1.00 -0.78 -1.22 -0.89 -1.47 -0.92 0.50 -0.11 0.01 -0.22 0.02 -0.33 0.00 -0.57 -0.03 0.75 0.58 0.70 0.45 0.69 0.56 0.89 0.28 0.83 0.90 1.18 1.30 1.00 1.23 1.30 1.64 1.03 1.57 0.95 1.54 1.65 1.30 1.53 1.76 2.09 1.46 1.99

d ll=O·2 0.05 -2.15 -2.04 -2.68 -2.44 -2.62 -2.25 -3.28 -2.34 0.10 -1.71 -1.58 -2.15 -1.87 -2.11 -1.73 -2.73 -1.83 0.25 -0.93 -0.81 -1.28 -0.94 -1.30 -0.91 -1.88 -0.98 0.50 -0.17 -0.02 -0.43 -0.05 -0.41 -0.02 -0.97 -0.11 0.75 0.54 0.69 0.29 0.68 0.45 0.85 -0.05 0.76 0.90 1.13 1.28 0.77 1.16 1.24 1.63 0.70 1.50 0.95 1.44 1.60 0.98 1.42 1.69 2.08 1.13 1.94

d=l11 0.05 -2.36 -2.23 -3.17 -3.09 -2.76 -2.30 -3.82 -2.62 0.10 -1.83 -1.70 -2.58 -2.46 -2.27 -1.79 -3.26 -2.04 0.25 -1.09 -0.90 -1.68 -1.42 -1.44 -0.94 -2.35 -1.13 0.50 -0.25 -0.05 -0.78 -0.28 -0.56 -0.05 -1.37 -0.17 0.75 0.46 0.68 0.01 0.58 0.32 0.84 -0.43 0.71 0.90 0.98 1.22 0.50 1.06 1.09 1.59 0.35 1.44 0.95 1.28 1.50 0.70 1.35 1.51 2.03 0.76 1.86

10,000 ~lications. N=100 a2:1. The 5tl), Oth, 25th, 50th, 75th, 90th, and 95th quantiles for the standard nonnal distribution are, respectlvely, -1.64, -1.28, -0.67, O, 0.67, 1.28 ano 1.64.

rabIe 6 (continued)

Employment equations SNM and GMM estimates fmm tite UK sample

Dependent Sample perlod: 1979-1984 (140 companies) variable: &lit

AR(2) Models Independent variables GMM SNM GMM SNM

0.691 1.635 0.320 0.827 (0.051) (0.074) (0.053) (0.065) -0.114 -0.439 0.022 -0.094 (0.026) (0.039) (0.022) (0.032) 0.598 1.958

(0.070) (0.095) 0.0l3 -0.075

(0.036) (0.053)

Sargan test (df) 65.9 (50) 71.3 (50) 32.8 (25) 31.3 (25)

R2 's for IVs: &li(t.l) 0.216 0.152

Notes to Table 6

(i) Time durnmies are inc\uded in all equations. (ii) Asyrnptotic standard errors robust to heteroskedasticity are reported in parentheses. (iii) AH reported estimates are two step. (iv) Model A treats &!¡(I.I) , 6,wil, 6,Wi(I.I)' and&¡! as endogenous. Model B treats &!¡(I.l) ,6,Wi(l.l), and &¡ l.) as endogenous. (v)The instrument set for Models A and B includes lags ofemployrnent dated (t-2) and earlier, lags ofwages and capital dated (t-2) and (t-3) and the levels and first differences offirm real sales and firm real stocks dated (t-2). The instrument set for all the AR(2) models inc\udes lags ofemployrnent dated (t-2) and earlier, and for those in the first two columns also lags of wages dated (t-2) and earlier. (vi) The R2 's for the IVs denote the partial R2 between the instruments and each endogenous explanatol)' variable once the exogenous variables inc\uded in the equation have been partialled out.

rabIe 7

VAR estimates for employment and wage equations fmm the Spanish sample

Sample perlad: 1986-1990 (738 companies)

Independent variables

3I1u Equation

.óni(t-2)

Sargan test (df)

R2 's for IVs: .óni(t-I)

~Wi(t.l)

Alvu Equation

Sargan test (df)

R2 '5 for IVs:

~Wi(t-I)

"MadeI 1" restrictions

GMM SNM GMM SNM

0.842 1.087 0.748 0.812 (0.669;1.015) (0.894;1.280) (0.575;0.921) (0.636;0.988) [0.470;1.004] [0.729;1.258] [0.505;0.989] [0.541;0.995]

-0.003 -0.074 0.038 0.030 (-0.060;0.054) (-0.140;-0.008) (-0.005;0.081) (-0.015;0.075) [-0.030;0.137] [-0.110;0.067] [-0.012;0.113] [-0.015;0.113]

0.078 0.222 (-0.086;0.242) (0.046;0.398) [-0.299;0.199] [-0.183;0.377]

-0.053 -0.074 (-0.102;-0.004) (-0.127;-0.021) [-0.110;0.021] [-0.137;-0.003]

36.9 (36) 37.2 (36) 14.4 (18) 13.5 (18)

0.033 0.031

0.178 0.228 0.178 0.228 (-0.042;0.398) (-0.008;0.464) (-0.042;0.398) (-0.008;0.464) [-0.170;0.491] [-0.172;0.636] [-0.208;0.542] [-0.237;0.734]

-0.012 -0.002 -0.012 -0.002 (-0.081 ;0.049) (-0.066;0.062) (-0.081;0.049) (-0.066;0.062) [-0.082;0.073] [-0.076;0.101] [-0.082;0.082] [-0.078;0.108]

12.7 (18) 12.9 (18) 12.7 (18) 12.9 (18)

0.019

Iabl~ 6 EmpIoyment equations

SNM and GMM estimates fmm 1he UK sample

I ¡

De1?':ndent varIable: LIDit

Independent variables

LIDi(t-l)

LIDi(t-2)

~Wit

~Wi(t-l)

&¡t

&¡(t-I)

~ySit

~ySí(t-l)

~ySi(t-2)

Sargan test (df)

R2 'S fQ[ IVs: LIDí(t-l) ~Wit ~Wi(t-l) &¡t

&¡(t-I)

lDependent variable is L\wit•

Sample period: 1979-1984 (140 companies)

Model A Model B

Indirect GMM SNM GMM1 GMM SNM

0.800 1.596 1.214 0.825 2.186 (0.048) (0.105) (0.056) (0.216) -0.116 -0.384 -0.282 -0.074 -0.455 (0.021) (0.045) (0.020) (0.077) -0.640 -1.897 -4.638 (0.054) (0.160) 0.564 2.138 1.567 0.431 2.841

(0.066) (0.142) (0.076) (0.312) 0.219 0.238 0.604

(0.051) (0.089) -0.077 -0.787 (0.045) (0.126)

0.890 1.747 3.105 (0.098) (0.204) -0.874 -2.897 -4.101 -0.115 -2.438 (0.105) (0.229) (0.100) (0.358)

0.095 1.511 (0.091) (0.266)

63.0 (50) 67.1 (50) 62.8 (50) 68.3 (51) 66.5 (51)

0.271 0.269 0.193 0.309 0.289 0.108

0.158

Table 7 (continued)

VAR estimates for employment and wage equations fmm the Spanish sample

Sample perlod: 1986-1990 (738 companies)

"Model 3" restrictions Independent variables GMM SNM

&íj¡ Equation

1.163 1.208 (1.112;1.214) (1.137;1.279) [1.064;1.222] [1.157;1.370]

llni(t.2) -0.135 -0.142 (-0.172;-0.098) (-0.185;-0.099) [-0.166;-0.044] [-0.178;-0.033]

0.121 . 0.116 (0.086;0.156) (0.077;0.155) [0.075;0.166] [0.054;0.154]

-0.132 -0.151 (-0.171;-0.093) (-0.194;-0.108) [-0.180;-0.073] [-0.232;-0.113]

Sargan test (df) 80.1 (48) 69.1 (48)

Alvit Equation

0.854 0.873 (0.815;0.893) (0.834;0.912) [0.790;0.888] [0.828;0.926]

0.152 0.138 (0.105;0.199) (0.089;0.187) [0.107;0.235] [0.074;0.207]

Sargan test (df) 71.4 (24) 72.2 (24)

Notes lo Table 7

(i) Time durnmies are included in all equations. (ii) AH reported estimates are two step. (iii) The instrument set for all the employment equations under "Model 1" includes lags ofemployment dated (t-2) and earlier, and for those in the ftrSt two columns also lags of wages dated (t-2) and earlier. The instrument set for the wage equation under "Model 1" includes lags of wages dated (t-2) and earlier. (iv) The R2 's for the IVs denote the partial R2 between the instruments and each endogenous explanatory variable once the exogenous variables included in the equation have been partialled out. (v) 95% asymptotic confidence intervals based on heteroskedasticity-robust standard errors in parentheses; 95% moment-restricted bootstrap confidence intervals in brackets. (vi) The bootstrap confidence intervals under "Model 1" for the equations in the ftrSt two columns are based on a distribution that satisfies a larger set of moment conditions than those in the third and fourth columns. The reason is that the former include lagged wages as instruments for the employment equation, which are absent from the latter.

Table 8

VAR estimares for employment and wage equatiom fmm the Spanish sample

Random sample containing 200 companies

Sample period: 1986-1990 (200 companies)

Independent variables GMM SNM GMM SNM

0.788 1.160 0.441 0.815 (0.610;0.966) (0.888;1.432) (0.167;0.715) (0.509;1.121) [0.037; 1.234] [0.365; 1.657] [-0.609;0.812] [0.237;1.566]

-0.042 -0.206 0.063 0.003 (-0.109;0.025) (-0.306;-0.106) (0.002;0.124) (-0.062;0.069) [-0.101;0.235] [-0.370;0.138] [0.000;0.221] [-0.109;0.145]

0.337 0.650 (0.151;0.523) (0.371;0.929) [-0.238;0.950] [0.090; 1.759]

0.001 -0.040 (-0.065;0.067) (-0.120;0.040) [-0.098;0.290] [-0.108;0.254]

Sargan test (df) 30.2 (36) 23.0 (36) 23.3 (18) 24.3 (18)

R2 '8 for IV8: LIDi(t.l) 0.064 ~w¡(t-1) 0.080

.&vil Equation

-0.612 -1.198 -0.612 -1.198 (-0.984;-0.240) (-1.442;-0.953) (-0.984;-0.240) (-1.442;-0.953) [-3.837;0.314] [-4.183;-0.933] [-3.766;0.227] [-3.989;-0.893]

-0.120 -0.270 -0.120 -0.270 (-0.231;-0.009) (-0.349;-0.l91) (-0.231;-0.009) (-0.349;-0.l91) [-0.715;0.1 07] [-0.878;-0.183] [-0.840;0.067] [-0.958;-0.160]

Sargan test (df) 17.3 (18) 11.0 (18) 17.3 (18) 11.0 (18)

R2 '8 for IVs: ~Wi(t.l) 0.023

See Notes to Table 7.

t !

Iabl~ 2 MOnte Carlo simulations for the AR(2) model for employment

a¡=tl.813, ~ =tl.03, y=(J.777, d TJ=tl.038, d v=tl.01

N=738 N=200

GMM SNM GMM SNM

SUll1lllalY of estimates

a¡ median 0.72 0.82 0.56 0.82 % bias 11.6 0.8 30.8 1.1

iqr 0.15 0.15 0.26 0.27 iq80 0.28 0.31 0.53 0.58 mae 0.11 0.08 0.25 0.14

median 0.01 0.03 -0.02 0.02 % bias 57.7 5.9 165.6 33.8

iqr 0.03 0.04 0.06 0.08 iq80 0.07 0.07 0.11 0.14 mae 0.02 0.02 0.05 0.04

Quantiles of the t-rntios

a¡ 0.10 -2.41 -1.40 -3.43 -1.63 0.25 -1.75 -0.71 -2.66 -0.76 0.50 -0.98 0.06 -1.82 0.06 0.75 -0.20 0.77 -0.96 0.85 0.90 0.47 1.40 -0.20 1.43

0.10 -2.13 -1.47 -2.85 -1.87 0.25 -1.37 -0.80 -2.07 -1.13 0.50 -0.70 -0.07 -1.26 -0.23 0.75 0.05 0.71 -0.43 0.60 0.90 0.67 1.30 0.26 1.25

1,000 replications. % bias gives the percentage median bias for al! estimates; iqr is the 75th-25th interquartile range; iq80 is the 90th-10th interquantile range; mae denotes the median absolute error. The 10th, 25th, 50th, 75th and 90th quantiles for the standard normal distribution are, respectively, -1.28, -0.67, O, 0.67 and 1.28.